Environmental Engineering Reference
In-Depth Information
distance d
x
. Moreover, assume that the surface width
B
s
is constant (pris-
matic section) and that the water is incompressible. For the unsteady flow
conditions the discharge
Q
changes with distance d
x
, namely d
Q
/d
x
and
the water depth
y
changes with time: d
y
/d
t
. The change of discharge
Q
through space in a small time-step d
t
is (d
Q
/d
x
)d
x
d
t
. The corresponding
change in storage in space is
B
s
d
y
/d
t
d
x
d
t
. From continuity considerations
(volumetric balance) follows that a change of discharge
Q
in
x
-direction
must be accompanied by a change in water depth
y
in time step
t
:
d
Q
d
x
d
x
d
t
B
s
d
y
+
d
t
d
x
d
t
=
0
∂Q
∂x
+
B
s
∂y
∂t
=
0
Continuity equation for unsteady flow
(2.43)
From this equation follows that the change in discharge
Q
in the
x
-direction is opposite to the change in water depth
y
during time step
t
.
Assuming that there is no lateral discharge (no inflow or outflow:
q
=
0) in
x
and with
Q
=
vA
·
∂Q
∂x
=
∂A
v
∂x
=
v
∂A
A
∂v
∂x
∂x
+
A
∂v
v
∂A
B
s
∂y
Continuity equation for a channel
with a general shape
∂x
+
∂x
+
∂t
=
0
(2.44)
The first term on the left-hand side represents the prism storage, the
second term represents the wedge storage and
B
s
∂y/∂t
is the rate of
rise of the water level with time
t
.
Prism storage
+
wedge storage
+
rate of rise of water level
=
0
With the hydraulic depth
D
=
A/B
s
and with d
A
=
B
s
d
y
, the continuity
equation becomes:
D
∂v
v
∂y
∂y
∂t
=
Continuity equation for a channel
with a general shape
∂x
+
∂x
+
0
(2.45)
For a rectangular channel with
q
=
Q/B
,
v
=
q/y
and
A
=
B
s
y
follows:
y
∂v
v
∂y
∂y
∂t
=
Continuity equation for a
rectangular channel
∂x
+
∂x
+
0
(2.46)
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